Transcript Διαφάνεια 1

Developing ideas of
refraction, lenses and
rainbow through the use of
Medieval Resources of Arabs
and Persians
Pavlos Mihas
Democritus University
Learning to apply a law
Usually students do not learn to apply general
principles.
In optical phenomena there is a need to learn
to apply at least qualitatively Snell’s law.
There are some useful approximations:
 Linear approximation
2
angle of refraction  angle of
3
This approximation holds up to 40°
incidence
Quadratic approximation
refraction angle  0,0025  incidence angle ²  b  incidence angle
Usefulness of a law
It is good to have an idea of what are the
values expected.
This helps the students to repeat a
measurement.
A comparison with Historical
measurements also helps to see how well
they perform
So we do measurements and compare
with “law – like” relations and historical
data.
Historical data and methods
 Ptolemy’s method: A
semicircle and 3 pins
 At first we put a pin B at
the center and then we
choose the angle of
incidence where we put
A. A pin C is moved
along the curved side
until the 3 pins overlap.
 Usually it works well up
to 80°.
Extensions of Ptolemy’s law
 The same idea can
be applied to other
shapes
Which Law?
PTOLEMY’S RESULTS FOR
REFRACTION FROM AIR TO
WATER
Angle of
Incidence
Angle of
Refraction
0°
0°
Increase in Difference
the Angle of of Increase
Refraction
8°00’
10°
8°
30’
PTOLEMY’S
RESULTS
It seems
that Ptolemy
gaveFOR
results that
REFRACTION
FROM AIR
have
constant Differences
of TO
the
Increase of theGLASS
Refraction Angle
Angle of
Incidence
Angle of
Refraction
Does
0° this hold
0° also for his air to glass
7°00’
data?
10°
7°
15°30’
30’
20°
13°30’
22°30’
30’
30°
19°30’
29°
30’
40°
25°
35°
30’
50°
30°
40°30’
30’
60°
34°30’
45°30;
30’
70°
38°30;
50°
30’
3°30’
4°30’
80°
30’
4°00’
5°00’
70°
30’
4°30’
5°30’
60°
30’
5°00’
6°00’
50°
30’
5°30’
6°30’
40°
30’
6°00’
7°00’
30°
30’
6°30’
7°30’
20°
Increase in Difference
the Angle of of Increase
Refraction
80°
42°
Ptolemy’s results and quadratic law
60
REFRACTION ANGLE
50
40
y = -0,0025x 2 + 0,725x
R2 = 1
GLASS
y = -0,0025x 2 + 0,825x
R2 = 1
WATER
Angle of Refraction
WATER
Angle of Refraction
GLASS
30
Quadratic (GLASS)
Quadratic (WATER)
20
10
0
0,00
20,00
40,00
60,00
INCIDENCE ANGLE
s
80,00
Students
measurements
can be very well
fit in a quadratic
law
Why we have to believe Snell’s
Law?
 At this point we can stress the importance
of theory in experiments
 Scientists design their experiments
according to the theory they hold.
 The data they collect reflect the theory or
Law they believe.
 Ptolemy believed in “visual Rays” ὄψεις,
while Al Haytham ‫ الهىثام‬believed that light
comes to our eyes from the bodies.
Al Haytham’s method
 Al Haytham employed a
light beam. He used two
diaphragms (UHF, hole
on MPQN) to make a
“parallel beam”
 He measured the angle
of deviation KCI
 He did not give one
general relation but
some rules.
 His rules are not
accurate for ranges that
exceed Ptolemy’s
results.
 Students can check Al
Haytham’s rules
Checking Al Haytham’s rules
•In rare to dense refraction, deviation angle<1/2 incidence angle
Interpretation of the Law
 Corpuscular Interpretation
 Wave Interpretation
Al Haytham’s interpretation
• Al Haytham proposed a corpuscular
interpretation
• He expressed the idea of vector Analysis:
There is a portion (‫ قسط‬koust) of velocity in
a direction parallel to the surface and a
portion perpendicular to the surface .
• According to Al Haytham, light must be
deviated toward the path of least
resistance.
• The resistance is smallest in the direction
perpendicular to the surface.
• If refraction is a weaker case of reflection,
why refraction is not away from the
perpendicular?
Newton’s Interpretation
• A kind of gravitational
field exists.
• This field accelerates
the particles when they
enter in a “denser”
medium, and
decelerates them when
they enter into a rarer
medium.
• When the particles exit
then they are
decelerated.
This interpretation avoids all the difficulties that were
presented by Al Haytham’s treatment. It can be used to
describe the behavior of lenses and prisms.
Which model to teach?
• In a 8th grade Greek textbook is
presented Fermat’s principle.
• In a 7th grade Albanian textbook is
presented a corpuscular theory.
Particle model for reflection in Prifti et al (2003):
elastic ball on the left impinging on the floor and
light on the right impinging on a mirror
Which model for refraction?
• It is inconsistent to teach refraction
with a particle model
Refraction of sound in water (air in the upper part) (Prifti et
al 2003). On the right is shown a wave refraction
Wave model vs. corpuscular
model
• Wave model is easier to understand and
gives a grasp to the interested students.
• Corpuscular model is misleading. It directs
the student’s thought to classical bodies. For
example the explanation of the shadow in
comparison to rain or dust.
• Photon model should be taught because of
quantum mechanics, but the ideas of
quantum probabilities and Feynman’s path
integrals are not appropriate for students of
7th of 8th grade.
Refraction and the problem
of focusing
• Ibn Sahl’s law of refraction
Ibn Sahl's expression of Snell's law
Ibn Sahl’s perfect focusing
Ibn Sahl’s Hyperbolic Lens
Refraction on a Sphere: Special rays
If we consider the point where the ray meets the sphere after refraction we can see
A that there is a limit for a special angle
of incidence. Rays with a larger incidence
angle meet the sphere at a point nearer to
the axis.
The PARALLEL rays cut the SPECIAL ray
either inside the sphere or outside.
Geometrically we have
 sin(  i ) 
 sin(  c ) 
arcsin

arcsin



)
lim(
 n 
 n 
 i  c

Relation of rays to the special ray
of critical angle
1
 i   c 
lim
2 i  c
• The ray that Al Haytham
proposed as a limiting case
was employed by Al Farisi to
divide the incident rays to the
exterior and interior cone.
Refraction and the Rainbow
External and Internal cones of Al Farisi
Refraction and Observations
• Al Haytham’s Theory about the
influence of atmospheric refraction
on the observation of astronomical
phenomena
• Al Haytham proposed that the
atmosphere causes a refraction of
the light. This causes a change in
the angle where the Moon is seen.
Al Haytham’s model
• He proposed several factors for the
change of the appearance:
• Refraction by air
• Possibility of existence of a layer of
vapors over the atmosphere.
• Possibility of an “error of
observation”
What model for the
atmosphere:
Atmosphere: Vapors